| Nonlinear functional analysis as one of the basic tools in researching manynonlinear problems, is widely used in dealing with nonlinear differential equations.Nonlinear ordinary differential boundary value problem as an important part of thenonlinear differential equation problems has important application value andtheoretical significance.Nonlocal problem of ordinary differential equations is that the boundarycondition is not only dependent on the solution values at the interval endpoints, butrelies on the solution values of some points in its interior. Moreover, two-pointboundary value problems and three-point boundary value problems have become theresearch focus since they have rich applied backgrounds.Fractional calculus concepts have a long history, but not been valued enough.Until recent decades they have been more and more used in many applied fields,such as fractal, thermal system, optics, materials, information processing, robot andcontrol, system identification, and so on, so that the fractional calculus has becomeone of the hot research spots. The fractional model is more accurate than the integerorder model, because the fractional model has more degrees of freedom. Thisdissertation is divided into the following five chapters.Chapter1mainly introduces backgrounds of selected topics and issues, theinnovation of the thesis.Chapter2mainly studies and proves the existence of positive solutions for theSturm-Liouville boundary value problems of a class of two order ordinary differentialequations.Chapter3proves the existence of multiple solutions for a class of three pointboundary value problems and some related theorems.Chapter4discusses the existence of solutions for a class of fractional differentialequations and some relevant results, and then gives a concrete example.Chapter5studies the numerical solutions for a kind of the space-time fractionalconvection diffusion equations, and includes difference scheme, stability, and proof ofconvergence. |
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